We essentially use the arguments from [1]} and [2]}. Let \(\lbrace (E_j,d_j)\rbrace _{1\le j \le m}\)  be a family of separable metric space each endowed with a finite doubling measure \(\upsilon _j\)  defined on the Borel sets of \(E_j\)  (we essentially need that Lebesgue's differentiation theorem holds) and \(\Psi \)  a positive increasing convex function such that \(\Psi (0) = 0\)  (and, thus, \(\Psi ^{-1}\)  is a positive increasing concave function). We define \(\sigma _j(r) = \min _{x\in E_j} \upsilon _j(B(x,r))\)  the volume (under \(\upsilon _j\) ) of the smallest ball of radius \(r>0\)  in \(E_j\) . For any metric space \((E,d)\)  endowed with a measure \(\upsilon \)  defined on the Borel sets of \(E\)  and for any scalar-valued function \(f\in \mathcal {C}(E)\) , we define
\(\bar{f}(A) : = \upsilon (A)^{-1} \int _{A} f(x) \upsilon (dx),\) 
for any Borel set \(A\in E\) . For any \((E,d,\upsilon )\in \lbrace (E_j,d_j,\upsilon _j)\rbrace _{1\le j\le m}\) , we have, by Lebesgue's differentiation theorem
\(\bar{f}(B(x,r)) \underset{r\rightarrow 0}{\rightarrow } f(x).\) 
We denote in the following \(\mathcal {E}= \bigotimes _{j = 1}^m E_j\)  the product space that is can be made metric by considering \(d(x,y) = \left(\sum _{j=1}^m d(x_j,y_j)^2\right)^{1/2}\) . It can also be made measurable by considering the product measure \(\upsilon (A) = \prod _{j = 1}^m \upsilon (A_j)\)  for any Borel set \(A = \prod _{j = 1}^m A_j\) . For any \(x = (x_1,x_2,\ldots ,x_m),y = (y_1,y_2,\ldots ,y_m)\in \mathcal {E}\) , we denote \(x^{\prime } = (x_1,\ldots ,x_{m-1})\)  (for \(m\ge 2\) ) and define, for any \(1\le k\le m\) ,
\(V_{k,y}x : = (x_1,\ldots ,x_{k-1},y_k,x_{k+1},\ldots ,x_m).\) 
Furthermore, for any function \(f\)  defined on \(\mathcal {E}\) , we denote \(V_{k,y}f(x) = f(V_{k,y}x)\) . We proceed to define the joint increment of \(f\)  as
\(\square _y f(x) : = \prod _{j = 1}^m (\textrm {id} - V_{k,y})f(x).\) 
From the previous definitions, we can see that, for any \(x,y\in \mathcal {E}\)  and \(f\)  defined on \(\mathcal {E}\)
\(\square _y f(x) = \square _{y^{\prime }} f(x^{\prime },x_m) - \square _{y^{\prime }} f(x^{\prime },y_m).\) 
We can now state an intermediate result.
Lemme 4.3 
Let \(f\in \mathcal {C}(\mathcal {E})\)  be such that
\(\kappa _f : = \int _{\mathcal {E}}\int _{\mathcal {E}} \Psi \left(\frac{|\square _yf(x)|}{\prod _{j=1}^m d_j(x_i,y_j)} \right)\upsilon (dx)\upsilon (dy)< +\infty .\) 
Then the following inequality holds for any \(x,y\in \mathcal {E}\)
\(|\square _yf(x)| \le 18^m \int _0^{d_1(x_1,y_1)/2}\ldots \int _0^{d_m(x_m,y_m)/2}\Psi ^{-1}\left(\frac{\kappa _f}{\prod _{j = 1}^m \sigma _j(r_j)^2}\right)\mathop {}\!\mathrm {d}r_1\ldots \mathop {}\!\mathrm {d}r_m.\) 

We proceed by induction; the case \(m = 1\)  being the classical Garsia-Rodemich-Rumsey. We assume that the result holds for a certain \(m-1\ge 1\) . For any \(x,y\in \mathcal {E}\) , we denote \(g_{x^{\prime },y^{\prime }}(z) =\square _{y^{\prime }} f(x^{\prime },z)\) . For any Borel sets \(A_1,A_2\subset E_m\) , we have
\(\left|\bar{g}_{x^{\prime },y^{\prime }}(A_1) - \bar{g}_{x^{\prime },y^{\prime }}(A_2)\right| &\le \int _{A_1}\int _{A_2} \left|g_{x^{\prime },y^{\prime }}(x_m) - g_{x^{\prime },y^{\prime }}(y_m)\right|\frac{\upsilon _m(dx_m)\upsilon _m(dy_m)}{\upsilon _m(A_1)\upsilon _m(A_2)}\\ &\le d_m(A_1,A_2) \int _{A_1}\int _{A_2} \left|\frac{g_{x^{\prime },y^{\prime }}(x_m) - g_{x^{\prime },y^{\prime }}(y_m)}{d_m(x_m,y_m)}\right|\frac{\upsilon _m(dx_m)\upsilon _m(dy_m)}{\upsilon _m(A_1)\upsilon _m(A_2)},\) 

where \(d_m(A_1,A_2) = \sup _{x_m\in A_1,y_m\in A_2} d_m(x_m,y_m)\) .
By the induction hypothesis, we have that
\(\frac{|g_{x^{\prime },y^{\prime }}(x_m) - g_{x^{\prime },y^{\prime }}(y_m)|}{d_m(x_m,y_m)} = \left|\square _{y^{\prime }} \delta _{x_m,y_m}f(x^{\prime })\right|\\ \le 18^{m-1} \int _{D}\Psi ^{-1}\left(\frac{\kappa _{\delta _{x_m,y_m}f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\right)\mathop {}\!\mathrm {d}r,\) 

where we denote \(D = [0,d_1(x_1,y_1)/2]\times \ldots \times [0,d_{m-1}(x_{m-1},y_{m-1})/2]\) , \(r = (r_1,\ldots ,r_{m-1})\)  and \(\delta _{x_m,y_m}f(x^{\prime }) = (f(x^{\prime },x_m) - f(x^{\prime },y_m))/d_m(x_m,y_m)\) . Thus, by Fubini's theorem and Jensen's inequality, we obtain that
\(\left|\bar{g}_{x^{\prime },y^{\prime }}(A_1) - \bar{g}_{x^{\prime },y^{\prime }}(A_2)\right|\\ \le 18^{m-1}d_m(A_1,A_2)\int _{D} \int _{A_1}\int _{A_2} \Psi ^{-1}\left(\frac{\kappa _{\delta _{x_m,y_m}f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\right)\frac{\upsilon _m(dx_m)\upsilon _m(dy_m)}{\upsilon _m(A)\upsilon _m(B)}\mathop {}\!\mathrm {d}r\\ \le 18^{m-1}d_m(A_1,A_2)\int _{D} \Psi ^{-1}\left(\int _{A_1}\int _{A_2} \frac{\kappa _{\delta _{x_m,y_m}f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{\upsilon _m(dx_m)\upsilon _m(dy_m)}{\upsilon _m(A_1)\upsilon _m(A_2)}\right)\mathop {}\!\mathrm {d}r.\) 

We remark that
\(\int _{A_1}\int _{A_2} \kappa _{\delta _{x_m,y_m}f}\upsilon _m(dx_m)\upsilon _m(dy_m) \le \int _{E_m}\int _{E_m} \kappa _{\delta _{x_m,y_m}f}\upsilon _m(dx_m)\upsilon _m(dy_m) = \kappa _f,\) 

and, thus, we obtain
\(\left|\bar{g}_{x^{\prime },y^{\prime }}(A_1) - \bar{g}_{x^{\prime },y^{\prime }}(A_2)\right| \le 18^{m-1}d_m(A_1,A_2)\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\upsilon _m(A_1)\upsilon _m(A_2)}\right)\mathop {}\!\mathrm {d}r.\) 
We now denote \(\bar{g}_{x^{\prime },y^{\prime }}(x_m,r) = \bar{g}_{x^{\prime },y^{\prime }}(B(x_m,r))\)  for any \(x_m\in E_m\)  and
\(B(x_m,r) = \lbrace  x\in E_m; d_m(x,x_m)\le r\rbrace \) 
for a certain \(r>0\) . We now fix \(x_m,y_m\in E_m\)  and let \(\lambda _k = d_m(x_m,y_m)2^{-k}\)  for any \(k\ge 0\) . Since \(d_m(B(x_m,\lambda _0),B(y_m,\lambda _0)) = 3 \lambda _0\) , \(\sigma _m(\lambda _k)\ge \sigma _m(\tau )\)  for any \(\tau \le \lambda _k\)  and \(\Psi ^{-1}\)  is increasing, it follows from (REF ) that
\(&\left|\bar{g}_{x^{\prime },y^{\prime }}(x_m,\lambda _0) - \bar{g}_{x^{\prime },y^{\prime }}(y_m,\lambda _0)\right| \nonumber \\ &\hspace{20.0pt}\le 3\times 18^{m-1}\lambda _0\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\lambda _0)^2}\right)\nonumber \\ &\hspace{20.0pt}\le 6\times 18^{m-1}\int _0^{d_m(x_m,y_m)/2}\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\tau )^2}\right)\mathop {}\!\mathrm {d}r \mathop {}\!\mathrm {d}\tau .\) 

Furthermore, by using (REF ) and since \(d_m(B(x_m,\lambda _\ell ),B(x_m,\lambda _{\ell +1})) = \lambda _{\ell } + \lambda _{\ell +1} = 6(\lambda _{\ell +1} - \lambda _{\ell +2})\)  for any \(\ell \ge 0\) , \(\sigma _m(\lambda _k)\ge \sigma _m(\tau )\)  for any \(\tau \le \lambda _k\)  and \(\Psi ^{-1}\)  is increasing, we obtain that, for any \(k\ge 1\) ,
\(&\left|\bar{g}_{x^{\prime },y^{\prime }}(x_m,\lambda _0) - \bar{g}_{x^{\prime },y^{\prime }}(x_m,\lambda _k)\right| \le \sum _{\ell = 0}^{k-1} \left|\bar{g}_{x^{\prime },y^{\prime }}(x_m,\lambda _{\ell +1}) - \bar{g}_{x^{\prime },y^{\prime }}(x_m,\lambda _{\ell })\right|\\ &\hspace{20.0pt}\le 6\times 18^{m-1}\sum _{\ell = 0}^{k-1}(\lambda _{\ell +1} - \lambda _{\ell +2})\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\lambda _{\ell +1})\sigma _m(\lambda _{\ell +2})}\right)\mathop {}\!\mathrm {d}r\\ &\hspace{20.0pt}\le 6\times 18^{m-1}\sum _{\ell = 0}^{k-1}\int _{\lambda _{\ell +2}}^{\lambda _{\ell +1}}\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\tau )^2)}\right)\mathop {}\!\mathrm {d}r \mathop {}\!\mathrm {d}\tau \\ &\hspace{20.0pt}\le 6\times 18^{m-1}\int _{\lambda _{k+1}}^{\lambda _{1}}\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\tau )^2)}\right)\mathop {}\!\mathrm {d}r\mathop {}\!\mathrm {d}\tau .\) 

Letting \(k\rightarrow +\infty \) , we deduce that
\(&\left|\bar{g}_{x^{\prime },y^{\prime }}(x_m,\lambda _0) - g_{x^{\prime },y^{\prime }}(x_m)\right| \le \nonumber \\ &\hspace{20.0pt}\le 6\times 18^{m-1}\int _{0}^{d_m(x_m,y_m)/2}\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\tau )^2)}\right)\mathop {}\!\mathrm {d}r\mathop {}\!\mathrm {d}\tau ,\) 

and, similarly,
\(&\left|\bar{g}_{x^{\prime },y^{\prime }}(y_m,\lambda _0) - g_{x^{\prime },y^{\prime }}(y_m)\right| \le \nonumber \\ &\hspace{20.0pt}\le 6\times 18^{m-1}\int _{0}^{d_m(x_m,y_m)/2}\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\tau )^2)}\right)\mathop {}\!\mathrm {d}r\mathop {}\!\mathrm {d}\tau .\) 

Thanks to (REF ), (REF ) and (REF ), we obtain the desired result.
Corollary REF  follows directly from Lemma REF  by choosing \(E_1 = [0,1]\)  endowed with \(d_1(t,s) = |t-s|^{\alpha _1+1/p}\)  and \(E_2 = B(0,R)\)  endowed with \(d_2(x,y) = |x-y|^{\alpha _2 +d/p}\) . We also use, for both, Lebesgue's measure and set \(\Psi (t) = |t|^{p}\) . It turns out that \(\sigma _1(r) \simeq r^{1/(\alpha _1+1/p)}\)  and \(\sigma _2(r) \simeq r^{1/(\alpha _2/d+1/p)}\) . With the previous choices at hand, we obtain
\(\int _0^{d_1(s,t)/2}\int _0^{d_2(x,y)/2}\Psi ^{-1}&\left(\frac{\kappa _f}{\sigma _1(r_1)^2\sigma _2(r_2)^2}\right)\mathop {}\!\mathrm {d}r_1\mathop {}\!\mathrm {d}r_2\\ &\simeq \kappa _f^{1/p} \int _0^{d_1(s,t)/2}r_1^{-2/(\alpha _1 p+1)}\mathop {}\!\mathrm {d}r_1\int _0^{d_2(x,y)/2} r_2^{-2/(\alpha _2 p/d+1)} \mathop {}\!\mathrm {d}r_2\\&\simeq \kappa _f^{1/p} |t-s|^{\alpha _1-1/p} |x-y|^{\alpha _2-d/p},\) 

where
\(\kappa _f = \int _{[0,1]^2}\int _{B(0,R)^2} \frac{|\square _{(s,x)}f(t,y)|^p}{|t-s|^{\alpha _1p +1}|x-y|^{\alpha _2 p +d}}dtdsdxdy.\) 
